Answer

Verified

428.1k+ views

**Hint**The question is asking us about the value of resistance so that the power dissipated in the rod is twice of that dissipated in the resistance, so, you simply need to find the value of resistance of the rod using the given relation of current and voltage and then, find the value of the resistance using the value of the resistance of rod and the relation of powers dissipated in both parts. Do remember that the current flowing through both the parts will be exactly the same as they are in series.

**Complete step by step answer:**

We will proceed with the solution exactly as explained in the hint section of the solution to the asked question. First, we will use the given relation between the current and voltage to find the value of the resistance of the rod, then we will use the value of the resistance of the metallic rod to find the value of the resistance in series with the help of the relation given between the powers dissipated in both parts.

Let us first have a look at what is given to us in the question for the metallic rod section:

$I = 0.2{V^{5/2}}$

Where $I$ is the current flowing through the metallic rod and,

$V$ is the potential difference across the metallic rod

We can also write it as:

\[I = \left( {\dfrac{1}{5}{V^{3/2}}} \right)V\] … (i)

If we represent the resistance of the metallic rod as $r$, we can write:

$r = \dfrac{V}{I}$

If we do a transpose in the equation (i), we can write:

\[

I = \left( {\dfrac{1}{5}{V^{3/2}}} \right)V \\

\Rightarrow \dfrac{V}{I} = \dfrac{5}{{{V^{3/2}}}} \\

\Rightarrow r = \dfrac{5}{{{V^{3/2}}}} \\

\]

Now, the question has told us that the power dissipated in the rod is twice of that of the power dissipated in the resistance in series, this can be mathematically represented as:

${P_M} = 2{P_R}$

Where, ${P_M}$ is the power dissipated in the metallic rod whilst,

${P_R}$ is the power dissipated in the series resistance part.

Let us assume the resistance in the series has a value of $R$

Since both the sections are in series, the current flowing through them will be the same, $I$

We can write both the powers as:

$

{P_M} = {I^2}r \\

{P_R} = {I^2}R \\

$

Substituting the values in the relation, we get:

$

{I^2}r = 2{I^2}R \\

\Rightarrow r = 2R \\

$

Putting in the value of \[r = \dfrac{5}{{{V^{3/2}}}}\] we get:

$R = \dfrac{5}{{2{V^{3/2}}}}$

Since both the resistances are in series, the net voltage in the circuit of $6V$ will be divided as:

$V = \dfrac{r}{{r + R}}6$

Substituting $r = 2R$, we get:

$V = \dfrac{{12R}}{{3R}} = 4V$

The value of $R$ then becomes:

$

R = \dfrac{5}{{2{V^{3/2}}}} \\

\Rightarrow R = \dfrac{5}{{2{{\left( 4 \right)}^{3/2}}}} \\

\Rightarrow R = \dfrac{5}{{16}} \\

\Rightarrow R = 0.3125\Omega \\

$

**Hence, we found out the value of the series resistance to be $R = 0.3125\Omega $**

**Note**A common mistake that many students make is missing out on the fact that current flowing through both the sections will be the same since they are in series connection and the manipulation of finding out the potential difference across the metallic rod.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How do you graph the function fx 4x class 9 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE